A New Construction of Isospectral Riemannian Nilmanifolds with Examples

Isometric actions on pseudo-Riemannian nilmanifolds

This work deals with the structure of the isometry group of pseudo-Riemannian 2-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannain situation; for instance the action of the nilradical of the isometry group does not need to be transitive. For a nilpotent Lie group endowed with left-invariant pseudo-Riemannian metric we study conditions for which the subgroup of isometries fixing the identity element equals the subgroup of isometric automorphisms. This set equality holds for pseudo-$H$-type groups.

测地轨道伪黎曼H型nilmanifold：最小可允许克利福德模块的情况

Geodesic orbit pseudo-Riemannian H-type nilmanifolds: case of minimal admissible Clifford modules

本文研究了伪黎曼H型nilmanifold的测地轨道性质，扩展了Riehm关于黎曼H型李群的研究，并针对由最小维度可允许克利福德模块构造的李代数提供了完整特征化。

Weakly Symmetric Pseudo-Riemannian Nilmanifolds

In an earlier paper we developed the classification of weakly symmetric pseudo--riemannian manifolds $G/H$ where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derived the classification from the cases where $G$ is compact. As a consequence we obtained the classification of semisimple weakly symmetric manifolds of Lorentz signature $(n-1,1)$ and trans--lorentzian signature $(n-2,2)$. Here we work out the classification of weakly symmetric pseudo--riemannian nilmanifolds $G/H$ from the classification for the case $G = N\rtimes H$ with $H$ compact and $N$ nilpotent. It turns o…

The Marked Length Spectrum vs. The Laplace Spectrum on Forms on Riemannian Nilmanifolds

The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the same marked length spectrum, they necessarily share the same Laplace spectrum on functions. In contrast, we present the first example of a pair of isospectral Riemannian manifolds with the same marked length spectrum but not the same spectrum on one-forms. Outside of the standard…

符号为$(n-2,2)$的伪黎曼测地轨道幂零流形

Pseudo-Riemannian geodesic orbit nilmanifolds of signature $\boldsymbol{(n-2,2)}$

本文将黎曼与Lorentz测地轨道幂零流形的结构定理推广到符号$(n-2,2)$的trans-Lorentz情形，按度量在换位子代数上是否退化分别刻画其幂零李代数。

A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold

We answer in the affirmative the question posed by Conti and Rossi on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ of signature $(3, 4)$ on a nilpotent Lie group of dimension 7, such that $g$ is Einstein and not Ricci-flat. We show that the pseudo-metric $g$ cannot be induced by any left-invariant closed $G_2^*$-structure on the Lie group. Moreover, some results on closed and harmonic $G_2^*$-structures on an arbitrary 7-manifold $M$ are given. In particular,…

Homogeneous geodesics in pseudo-Riemannian nilmanifolds

We study the geodesic orbit property for nilpotent Lie groups $N$ when endowed with a pseudo-Riemannian left-invariant metric. We consider this property with respect to different groups acting by isometries. When $N$ acts on itself by left-translations we show that it is a geodesic orbit space if and only if the metric is bi-invariant. Assuming $N$ is 2-step nilpotent and with non-degenerate center we give algebraic conditions on the Lie algebra $\mathfrak n$ of $N$ in order to verify that every geodesic is the orbit of a one-parameter subgroup of $N\rtimes\operatorname{Auto}(N)$. In addition …

Geodesic orbit and naturally reductive nilmanifolds associated with graphs

We study Riemannian nilmanifolds associated with graphs. We prove that such a nilmanifold is geodesic orbit if and only if it is naturally reductive if and only if its defining graph is the disjoint union of complete graphs and the left-invariant metric is generated by a certain naturally defined inner product.

Unitary Representations, $L^2$ Dolbeault Cohomology, and Weakly Symmetric Pseudo--Riemannian Nilmanifolds

We combine recent developments on weakly symmetric pseudo--riemannian nilmanifolds with with geometric methods for construction of unitary representations on square integrable Dolbeault cohomology spaces. This runs parallel to construction of discrete series representations on spaces of square integrable harmonic forms with values in holomorphic vector bundles over flag domains. Some special cases had been described by Satake in 1971 and the author in 1975. Here we develop a theory of pseudo--riemannian nilmanifolds of complex type and the nilmanifold versions of flag domains. We construct the…

签名 $(n-3,3)$ 的伪黎曼测地轨道幂零流形

Pseudo-Riemannian geodesic orbit nilmanifolds of signature $\boldsymbol{(n-3,3)}$

本文研究签名 $(n-3,3)$ 的伪黎曼测地轨道幂零流形，证明在导出代数上非退化时幂零步数限于 2 或 4，退化时给出双扩张约化，并用九维和八维例子说明结论的精确性。

Symmetric Killing tensors on nilmanifolds

We study left-invariant symmetric Killing 2-tensors on 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric, and construct genuine examples, which are not linear combinations of parallel tensors and symmetric products of Killing vector fields.

Isospectral towers of Riemannian Manifolds

In this paper we construct, for n >= 2, arbitrarily large families of infinite towers of compact, orientable Riemannian n-manifolds which are isospectral but not isometric at each stage. In dimensions two and three, the towers produced consist of hyperbolic 2-manifolds and hyperbolic 3-manifolds, and in these cases we show that the isospectral towers do not arise from Sunada's method.

洛伦兹自然约化幂零流形的对称性分布

The distribution of symmetry of Lorentzian naturally reductive nilmanifolds

本文研究了相对于某类等距传递子群自然约化的2步幂零洛伦兹李群，描述其迷向表示并证明其不动点产生对称性分布，推广了黎曼情形的已知结果。

Continuous families of isospectral Riemannian metrics which are not locally isometric

Two Riemannian manifolds are said to be isospectral if the associated Laplace-Belttrami operators have the same eigenvalue spectrum. If the manifolds have boundary, one specifies DIrichlet or Neumann isospectrality depending on the boundary conditions imposed on the eigenfunctions. We construct continuous families of (Neumann and Dirichlet) isospectral metrics which have different local geometry on manifolds with boundary in every dimension greater than 6 and also new examples of pairs of closed isospectral manifolds with different local geometry. These examples illustrate for the first time t…

Killing forms on $2$-step nilmanifolds

We study left-invariant Killing $k$-forms on simply connected $2$-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For $k=2,3$, we show that every left-invariant Killing $k$-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing $2$-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing $3$-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases,…

Isospectral Alexandrov Spaces

We construct the first non-trivial examples of compact non-isometric Alexandrov spaces which are isospectral with respect to the Laplacian and not isometric to Riemannian orbifolds. This construction generalizes independent earlier results by the authors based on Schueth's version of the torus method.

The Structure of Geodesic Orbit Lorentz Nilmanifolds

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifold…

Isospectral deformations of negatively curved Riemannian manifolds with boundary which are not locally isometric

To what extent does the eigenvalue spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold determine the geometry of the manifold? We give examples of isospectral manifolds with different local geometry including continuous families of isospectral negatively curved manifolds with boundary as well as various pairs of manifolds. The latter illustrate that the spectrum does not determine whether a manifold with boundary has negative curvature, whether it has constant Ricci curvature, and whether it has parallel curvature tensor, and the spectrum does not determine whether a clo…

非奇异弱对称nil流形

Non-singular weakly symmetric nilmanifolds

本文对单连通非奇异弱对称nil流形进行了完整分类，发现了具有3维中心的新族和14维单参数族，该分类基于非奇异2步幂零李群中每条测地线均为等距单参数群像的性质。

Isospectral deformations of closed Riemannian manifolds with different scalar curvature

We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds, more precisely, on $S^n\times T^m$, where $T^m$ is a torus of dimension $m\ge 2$ and $S^n$ is a sphere of dimension $n\ge 4$. These metrics are not locally homogeneous; in particular, the scalar curvature of each metric is nonconstant. For some of the deformations, the maximum scalar curvature changes during the deformation.

Positve Entropy Geodesic Flows on Nilmanifolds

Let T be the nilpotent group of 4 x 4 real upper triangular matrices. In this note we show that the Euler equations of certain left-invariant riemannian metrics on T have a horseshoe. We also show, with the aid of a numerical computation of a Melnikov-type integral, that the Euler equations of the sub-riemannian Carnot metric on T has a horseshoe. This sharpens an earlier result of Montgomery, Shapiro and Stolin who had shown that the equations are algebraically non-integrable.

Continuous families of isospectral metrics on simply connected manifolds

We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the first examples of simply connected Riemannian manifolds without boundary which are isospectral, but not isometric. For example, we construct continuous isospectral families of metrics on the product of spheres S^4\times S^3\times S^3. The metrics considered are not locally homogeneous. For a big class of such families, the set of critical values of the scalar cu…

Spectral geometry of Riemannian Legendre foliations

We obtain geometric characterizations of isospectral minimal Riemannian Legendre foliations on compact Sasakian manifolds of constant $ϕ$-sectional curvature.

测地轨道伪黎曼 $H$ 型幂零流形

Geodesic orbit pseudo-Riemannian $H$-type nilmanifolds

本文刻画了伪 $H$ 型幂零李群 $N_{r,s}$ 的测地轨道性质，证明仅 $(0,1)$、$(1,2)$ 及 $N_{3,4}$ 的最小模情形为 GO，并显式构造了后者的测地线生成群。

Spectral properties of 4-dimensional compact flat manifolds

We study the spectral properties of a large class of compact flat Riemannian manifolds of dimension 4, namely, those whose corresponding Bieberbach groups have the canonical lattice as translation lattice. By using the explicit expression of the heat trace of the Laplacian acting on $p$-forms, we determine all $p$-isospectral and $L$-isospectral pairs and we show that in this class of manifolds, isospectrality on functions and isospectrality on $p$-forms for all values of $p$ are equivalent to each other. The list shows for any $p$, $1 \le p \le 3$, many $p$-isospectral pairs that are not isos…

p-宽度的等谱问题：Zoll 度量的一个应用

The Isospectral Problem for p-widths: An Application of Zoll Metrics

本文提出p-宽度的等谱问题，探讨黎曼流形是否由其p-宽度唯一确定，并利用Zoll度量在S^2上构造反例证明其不成立。

Isospectral orbifolds with different maximal isotropy orders

We construct pairs of compact Riemannian orbifolds which are isospectral for the Laplace operator on functions such that the maximal isotropy order of singular points in one of the orbifolds is higher than in the other. In one type of examples, isospectrality arises from a version of the famous Sunada theorem which also implies isospectrality on $p$-forms; here the orbifolds are quotients of certain compact normal homogeneous spaces. In another type of examples, the orbifolds are quotients of Euclidean $\R^3$ and are shown to be isospectral on functions using dimension formulas for the eigensp…

Small Covers, infra-nilmanifolds and positive curvature

We show that all the small covers which are infra-nilmanifolds are exactly real Bott manifolds. This implies that any small cover which admits a flat Riemannian metric must be a real Bott manifold. In addition, we will study small covers which admit Riemannian metrics with positive or nonnegative Ricci curvature or sectional curvature. We will see that these geometric conditions put very strong restrictions on the topology of the small covers and the combinatorial structure of the underlying simple polytopes. Similar geometric problems are also studied for the real moment-angle complex of an a…

Isospectral metrics on five-dimensional spheres

We construct isospectral pairs of Riemannian metrics on S^5 and on B^6, thus lowering by three the dimension of spheres and balls on which such metrics have been constructed previously (S^{n\ge 8} and B^{n\ge 9}). We also construct continuous families of isospectral Riemannian metrics on S^7 and on B^8. In each of these examples, the metrics can be chosen equal to the standard metric outside certain subsets of arbitrarily small volume.

基于黎曼淹没和等谱轨道梯度流的特征值过程

Eigenvalue processes in light of Riemannian submersion and gradient flow of isospectral orbits

本文利用黎曼几何语言，证明了多种特征值过程是紧 Lie 群等距作用诱导的黎曼淹没投影结果，统一处理了不同设定下的特征值过程，并据此推导出特征向量过程和一般 $β>0$ 的 $β$ 过程。

Isospectral flat 3-manifolds

There is a well-known problem about isospectrality of Riemannian manifolds: whether isospectral manifolds are isometric. In this work we give an answer to this problem for 3-dimensional compact flat manifolds.

Higher degree Killing forms on $2-$step nilmanifolds

We study left-invariant Killing forms of arbitrary degree on simply connected $2-$step nilpotent Lie groups endowed with left-invariant Riemannian metrics, and classify them when the center of the group is at most two-dimensional.

On the Geometric Orbit Property for Lorentz Manifolds

The geodesic orbit property has been studied intensively for Riemannian manifolds. Geodesic orbit spaces are homogeneous and allow simplifications of many structural questions using the Lie algebra of the isometry group. Weakly symmetric Riemannian manifolds are geodesic orbit spaces. Here we define "naturally reductive" for pseudo-Riemannian manifolds and note that they are geodesic orbit spaces. A few years ago two of the authors proved that weakly symmetric pseudo-Riemannian manifolds are geodesic orbit spaces. In particular these results apply to pseudo-Riemannian Lorentz manifolds. There …

Topological Entropy of Left-Invariant Magnetic Flows on 2-Step Nilmanifolds

We consider magnetic flows on 2-step nilmanifolds $M = Γ\backslash G$, where the Riemannian metric $g$ and the magnetic field $σ$ are left-invariant. Our first result is that when $σ$ represents a rational cohomology class and its restriction to $\mathfrak{g} = T_eG$ vanishes on the derived algebra, then the associated magnetic flow has zero topological entropy. In particular, this is the case when $σ$ represents a rational cohomology class and is exact. Our second result is the construction of a magnetic field on a 2-step nilmanifold that has positive topological entropy for arbitrarily high …

On the isospectral orbifold-manifold problem for nonpositively curved locally symmetric spaces

An old problem asks whether a Riemannian manifold can be isospectral to a Riemannian orbifold with nontrivial singular set. In this short note we show that under the assumption of Schanuel's conjecture in transcendental number theory, this is impossible whenever the orbifold and manifold in question are length-commensurable compact locally symmetric spaces of nonpositive curvature associated to simple Lie groups.

Invariants of complex structures on nilmanifolds

Let $(N, J)$ be a simply connected $2n$-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on $N$ compatible with $J$ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. In [L1], J. Lauret proved that minimal metrics (if any) are unique up to isometry and scaling. This uniqueness allows us to distinguish two complex structures with Riemannian data, giving rise to a great deal of invariants. We show how to use a Riemannian invariant: t…

关于具有闭G2结构的二步nilmanifolds的nu不变量

On the nu-invariant of two-step nilmanifolds with closed G2-structure

研究了具有闭G2结构的二步nilmanifolds上的nu不变量，通过计算调和旋量和η不变量，确定了其存在性和奇偶性，并推导出在不变调和旋量上nu的消失。

Milnor's isospectral tori and harmonic maps

A well-known question asks whether the spectrum of the Laplacian on a Riemannian manifold $(M,g)$ determines the Riemannian metric $g$ up to isometry. A similar question is whether the energy spectrum of all harmonic maps from a given Riemannian manifold $(Σ,h)$ to $M$ determines the Riemannian metric on the target space. We consider this question in the case of harmonic maps between flat tori. In particular, we show that the two isospectral, non-isometric $16$-dimensional flat tori found by Milnor cannot be distinguished by the energy spectrum of harmonic maps from $d$-dimensional flat tori f…

Isospectral metrics on weighted projective spaces

We construct the first examples of families of bad Riemannian orbifolds which are isospectral with respect to the Laplacian but not isometric. In our case these are particular fixed weighted projective spaces equipped with isospectral metrics obtained by a generalization of Schüth's version of the torus method.

Equivariant isospectrality and Sunada's Method

We construct pairs and continuous families of isospectral yet locally non-isometric orbifolds via an equivariant version of Sunada's method. We also observe that if a good orbifold $\mathcal{O}$ and a smooth manifold $M$ are isospectral, then they cannot admit non-trivial finite Riemannian covers $M_1 \to \mathcal{O}$ and $M_2 \to M$ where $M_1$ and $M_2$ are isospectral manifolds.

通过广义 Heisenberg 群研究弱对称性与交换性的非紧致不可闻性

Non-compact inaudibility of weak symmetry and commutativity via generalized Heisenberg groups

本文构造一对 23 维广义 Heisenberg 群等谱流形，在非紧致情形下证明弱对称性和交换性均无法由谱信息识别。

Γ-extensions of the spectrum of an orbifold

We introduce the Γ-extension of the spectrum of the Laplacian of a Riemannian orbifold, where Γis a finitely generated discrete group. This extension, called the Γ-spectrum, is the union of the Laplace spectra of the Γ-sectors of the orbifold, and hence constitutes a Riemannian invariant that is directly related to the singular set of the orbifold. We compare the Γ-spectra of known examples of isospectral pairs and families of orbifolds and demonstrate that it many cases, isospectral orbifolds need not be Γ-isospectral. We additionally prove a version of Sunada's theorem that allows us to cons…

从三个视角看平坦环面的等谱问题

The isospectral problem for flat tori from three perspectives

本文从分析、几何和数论三个视角研究平坦环面的等谱非等距问题，回顾Milnor的16维例子，并介绍Schiemann在1990年代对最低存在维数问题的解答及若干未解决问题。

Ricci孤立子与爱因斯坦型流形的谱刚性

The spectral rigidity of Ricci soliton and Einstein-type manifolds

本文针对谱几何经典问题，在梯度收缩Ricci孤立子与上同调爱因斯坦流形两类条件下，证明除零散未知情形外，单个p对应的p-拉普拉斯特征值可确定截面曲率常值性，且可将等谱条件放宽为近等谱版本。

Torsion homology and regulators of isospectral manifolds

Given a finite group G, a G-covering of closed Riemannian manifolds, and a so-called G-relation, a construction of Sunada produces a pair of manifolds M_1 and M_2 that are strongly isospectral. Such manifolds have the same dimension and the same volume, and their rational homology groups are isomorphic. We investigate the relationship between their integral homology. The Cheeger-Mueller Theorem implies that a certain product of orders of torsion homology and of regulators for M_1 agrees with that for M_2. We exhibit a connection between the torsion in the integral homology of M_1 and M_2 on th…